X and Y Intercept Calculator
Type a function or an equation and get every point where it meets the axes — exactly where an exact answer exists, with the working shown, the holes ruled out, and each intercept marked on the graph.
2 x-intercepts, 1 y-intercept
Read as a function of x: the x-intercepts are its zeros and the y-intercept is its value at x = 0. Every value is exact.
- x-intercepts(2, 0), (3, 0)
- y-intercept(0, 6)
| x-intercept | At the axis the curve | Value |
|---|---|---|
| (2, 0) | crosses | exact |
| (3, 0) | crosses | exact |
How they were found
- y-intercept: put x = 0f(0) = 6, so the graph meets the y-axis at (0, 6).
- Solve for the x-interceptsx² − 5x + 6 is zero at x = 2, x = 3.
x² − 5x + 6 = 0
Functions and equations
Intercept questions come in two shapes, and the calculator takes both.
- A function, written
y = x² − 5x + 6,f(x) = …, or just the expression. The y-intercept is f(0) and the x-intercepts are the zeros of f. - An equation in x and y, like
2x + 3y = 6orx² + y² = 25. Put y = 0 and solve for the x-intercepts; put x = 0 and solve for the y-intercepts. A circle has two of each, which no function can.
Underneath they are the same idea. Every point on the x-axis has y = 0, so a point of the graph that lies on the x-axis is a solution with y = 0. Every point on the y-axis has x = 0, and the same reasoning applies.
One case is read literally on purpose. An equation with no y in it, such
as x = 3 or x² − 4 = 0, describes vertical
lines — every point whose x satisfies it, at any height. A vertical
line crosses the x-axis once and never meets the y-axis, so that is what
the calculator reports. This is also how any graphing tool draws those
equations. If you meant the graph of y = x² − 4, type the expression
on its own.
A zero on top is not always an intercept
A fraction is zero when its top is zero — provided its bottom is not. That proviso is where marks are lost.
y = (x² − 1)/(x − 1) has a numerator that is zero
at x = 1 and x = −1. But at x = 1 the denominator is zero too. The
factor (x − 1) cancels, and the graph is the straight
line y = x + 1 with a hole at (1, 2). The only x-intercept is
(−1, 0).
The sharpest version is y = (x − 1)²/(x − 1).
After cancelling it is y = x − 1, which is zero at x = 1 — so
the reduced formula says there is an x-intercept at (1, 0). There is not.
The original function is undefined at x = 1, so the graph is the line
y = x − 1 with the point (1, 0) missing: the hole sits exactly on the
axis, exactly where everyone would draw the intercept. The calculator
draws it hollow, lists it under “not intercepts, and why”, and
reports no x-intercept at all.
To get this right it looks at the numerator as you wrote it, checks every zero against the factors that cancel, and gives each one a verdict: an intercept, a hole, or a vertical asymptote.
Exact, not rounded
For polynomials and rational functions every intercept is exact.
x³ − 2x has intercepts at
−√2, 0 and √2. The circle
(x − 1)² + (y − 2)² = 9 meets the axes at
1 ± √5 and
2 ± 2√2.
That includes a case general equation solvers miss. A polynomial like
x⁴ − 5x² + 6 has no rational roots, so a solver that
only looks for rational roots before falling back to a numerical search
returns ±1.414214 and ±1.732051. But it factors as
(x² − 2)(x² − 3), and each quadratic piece has
exact roots. So the calculator factors completely first, using the same
engine as the factoring calculator,
and solves each piece: ±√2 and ±√3.
Only an irreducible piece of degree three or more is left to a numerical search — x³ − 3x + 1, whose three real roots cannot be written without cube roots of complex numbers — and those values are marked with ≈.
Crossing or touching
For each x-intercept of a function the calculator says whether the curve
crosses the axis there or touches it and turns back. That comes from how
many times the factor appears: (x − 2)²(x + 1) crosses
at x = −1 and touches at x = 2. A root repeated an even number of
times is touched; an odd number of times, crossed.
It matters for sketching, and it also explains a weakness of every numerical method: a search for roots looks for places where the function changes sign, and at a touching root it does not change sign. For polynomials and rational functions that does not matter here, because the roots are found exactly. For other functions it is stated alongside the result.
When there is no y-intercept
A function has at most one y-intercept, and sometimes none. The calculator says which reason applies:
- A vertical asymptote at x = 0, as with
1/x. The graph runs up the y-axis without ever touching it. - A hole at x = 0, as with
(x² + x)/x. After cancelling, the formula would give 1, but the point (0, 1) is missing, so the graph never quite reaches the y-axis. The hole is drawn on the graph. - Outside the domain, as with
ln x, which exists only for positive x.
What it does not do
- Intercepts in three dimensions, with a z-axis.
- Intercepts of a non-polynomial function outside the search window. A polynomial’s or a rational function’s are all found, wherever they are.
- Touching roots of a non-polynomial function, such as sin²x, which a search for sign changes cannot see.
- Intercepts read off a picture of a graph.
Questions
How do you find the x- and y-intercepts?
For the x-intercepts, set y = 0 and solve for x: they are the points where the graph meets the x-axis, and every point on that axis has y = 0. For the y-intercept, set x = 0 and work out y. For a function y = f(x) that means the y-intercept is simply f(0) and the x-intercepts are the zeros of f. The calculator shows both steps for whatever you type.
Why does it say x = 1 is not an intercept when the top is zero there?
Because the bottom is zero there too. In (x² − 1)/(x − 1), the factor (x − 1) cancels, and the graph is the line y = x + 1 with a single point missing at (1, 2). The numerator vanishing at a point where the function does not exist does not make an intercept. The same thing happens with (x − 1)²/(x − 1), where the missing point is right on the x-axis at (1, 0) — exactly where a dot would go — which is why the calculator draws it hollow and explains it rather than leaving it out silently.
Can a graph have more than one y-intercept?
A function cannot: it has one value at x = 0 at most, which is what the vertical line test says. But an equation in x and y can. The circle x² + y² = 25 meets the y-axis at (0, 5) and (0, −5). Type it as an equation and the calculator finds both.
What does "touches and turns back" mean?
That the curve meets the x-axis without crossing it. It happens at a repeated root: in (x − 2)²(x + 1) the factor (x − 2) appears twice, so the curve comes down to the axis at x = 2 and goes back up. A root that appears an odd number of times is crossed; one that appears an even number of times is touched. It matters when sketching, and it is also why a numerical search can miss such roots — there is no change of sign to find.
Why is the y-intercept missing for 1/x or ln x?
Because neither function has a value at x = 0. 1/x has a vertical asymptote there and ln x is only defined for positive x, so neither graph ever reaches the y-axis. The calculator says which reason applies rather than reporting an error.
Are the answers exact?
For polynomials, rational functions and equations built from them, yes: x⁴ − 5x² + 6 has intercepts at ±√2 and ±√3, written that way, not as ±1.414 and ±1.732. For anything involving sin, eˣ, ln and the rest, the zeros are found by searching a window, and values that are recognisably simple are named — sin x gives π, 2π and so on. Anything that is only approximate is marked with ≈.